Under ABA Model Rule 7.3(a), a message offering help with a particular matter, like a new award, to someone you know or reasonably should know needs that help is a solicitation. Live person-to-person contact (in person, live phone, real-time video) is barred when a significant motive is pecuniary gain, unless the person is a lawyer, has a family, close personal or prior business or professional relationship with the lawyer or firm, or routinely uses that type of legal service for business purposes. Written messages are still covered by Rules 7.1 and 7.3(c): nothing misleading, and nothing further once someone says they do not want to hear from you.
State rules differ. California has no business-purposes exception, bars real-time electronic contact as well as in-person and live phone contact, and requires the word 'Advertisement' on some written solicitations, so treat DMs and chat like live contact. Florida treats cold calls as prohibited solicitation. Its unsolicited emails must start the subject line with 'Advertisement', include a statement of the lawyer's background, training and experience with similar matters, say how the lawyer learned of any specific occurrence that prompted the message, and be filed with the Bar for review at least 20 days before use unless exempt.
Government contracts work adds its own traps. The Rule 7.1 comments say a truthful message still misleads if it causes a reasonable reader to think they must act when no action is required, so skip lines like 'Your award may be protested.' Rule 8.4(e) bars stating or implying an ability to improperly influence a government agency or official, which rules out any hint that a former agency lawyer at your firm can smooth things over with the contracting office. Rule 1.11 bars a former government lawyer from representing a client in a matter they participated in personally and substantially for the government, unless the agency gives informed consent, confirmed in writing.
Run conflicts first. If the firm represents a losing offeror or a protester on that procurement, do not pitch the awardee, and do not have a BD manager or agency make a call you could not make yourself. The same checks apply to any company you approach (see how to get corporate clients as a lawyer). None of this is legal or ethics advice; check your state's rules and, when in doubt, ethics counsel.